On asymptotic normality and variance estimation for nondifferentiable survey estimators

Abstract

Survey estimators of population quantities such as distribution functions and quantiles contain nondifferentiable functions of estimated quantities. The theoretical properties of such estimators are substantially more complicated to derive than those of differentiable estimators. In this article, we provide a unified framework for obtaining the asymptotic design-based properties of two common types of nondifferentiable estimators. Estimators of the first type have an explicit expression, while those of the second are defined only as the solution to estimating equations. We propose both analytical and replication-based design-consistent variance estimators for both cases, based on kernel regression. The practical behaviour of the variance estimators is demonstrated in a simulation experiment.